How to value a bond with a credit spread

Suppose I have a bond, with a face value of 95, a coupon of 2%, and a maturity of 50 years. Suppose the discount curve is flat at 2%, and there is no credit spread.

I am trying to calculate what happens when the credit/ interest rates change.

For example, how can I revalue the bond if the credit spread increases by 80bps, and interest rates increase to 3%?

Ideally I'd like to understand the theory behind it, as well as a method for calculating this (doesn't have to be exact, an approximation is fine).

In a world with flat discount curve at interest rate $$r$$ and flat default intensity $$\lambda$$ the bond recovery $$R$$ is related to the credit spread $$C$$ by $$\lambda=\frac{C}{1-R}\,.$$ If that bond has no market price and you are happy with those simplistic assumptions the dirty bond price is \begin{align} P&=\sum_{i=1}^n e^{-rt_i-\lambda t_i}K+e^{-rt_n-\lambda t_n}N+\int_0^{t_n}R\lambda e^{-(r+\lambda)s}\,ds\\ &=\sum_{i=1}^n e^{-rt_i-\lambda t_i}K+e^{-rt_n-\lambda t_n}N+R\lambda\frac{1-e^{-(r+\lambda)t_n}}{r+\lambda}\,. \end{align} Here $$N$$ is the face value and $$K$$ the cash amount of the bond coupon paid at time $$t_i\,.$$